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In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973.〔J.C. Harsanyi. 1973. "Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points. ''Int. J. Game Theory'' 2 (1973), pp. 1–23. 〕 The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent. The mixed strategy equilibria are explained as being the limit of pure strategy equilibria for a disturbed game of incomplete information in which the payoffs of each player are known to themselves but not their opponents. The idea is that the predicted mixed strategy of the original game emerge as ever improving approximations of a game that is not observed by the theorist who designed the original, idealized game. The apparently mixed nature of the strategy is actually just the result of each player playing a pure strategy with threshold values that depend on the ex-ante distribution over the continuum of payoffs that a player can have. As that continuum shrinks to zero, the players strategies converge to the predicted Nash equilibria of the original, unperturbed, complete information game. The result is also an important aspect of modern day inquiries in evolutionary game theory where the perturbed values are interpreted as distributions over types of players randomly paired in a population to play games. ==Example== Consider the Hawk-Dove game shown here. The game has two pure strategy equilibria (Defect, Cooperate) and (Cooperate, Defect). It also has a mixed equilibrium in which each player plays Cooperate with probability 2/3. Suppose that each player ''i'' bears an extra cost ''ai'' from playing Cooperate, which is uniformly distributed on (''A'' ). Players only know their own value of this cost. So this is a game of incomplete information which we can solve using Bayesian Nash equilibrium. The probability that ''ai'' ≤ ''a *'' is (''a *'' + ''A'')/2''A''. If player 2 Cooperates when ''a''2 ≤ ''a *'', then player 1's expected utility from Cooperating is - ''a1'' + 3(''a *'' + ''A'')/2''A'' + 2(1-(''a *'' + ''A'')/2''A''); his expected utility from Defecting is 4(''a *'' + ''A'')/2''A''. He should therefore himself Cooperate when ''a1'' ≤ 2 - 3(''a *''+''A'')/2''A''. Seeking a symmetric equilibrium where both players cooperate if ''ai'' ≤ ''a *'', we solve this for ''a *'' = 1/(2+3/''A''). Now we have worked out ''a *'', we can calculate the probability of each player playing Cooperate as : As A → 0, this approaches 2/3 - the same probability as in the mixed strategy in the complete information game. Thus, we can think of the mixed strategy equilibrium as the outcome of pure strategies followed by players who have a small amount of private information about their payoffs. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Purification theorem」の詳細全文を読む スポンサード リンク
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